Not just La Follette's progressivism but his opposition to US participation in World War I made him a pariah to the national Republican Party. La Follette got 24 votes at the 1920 Republican national convention--all from Wisconsin.
https://books.google.com/books?id=kNamDwAAQBAJ&pg=PT423&lpg=PT423 In 1924 he got 34 votes--24 from Wisconsin, 10 from North Dakota.
https://en.wikipedia.org/wiki/1924_Republican_National_Convention
The real question is whether he could have won as a third-party candidate. The answer is almost certainly No. If the economy was a little worse in 1924, he could have carried several western states which he narrowly lost in OTL (and where he far outperformed Davis). But see Walter Lippmann's comments ("Why I Shall Vote for Davis," in the
New Republic, October 29, 1924, reprinted in the chapter on the election of 1924 in Arthur M. Schlesinger, Jr., Fred L. Israel, and William P. Hansen, eds.,
History of American Presidential Elections, 1789-1968, Vol. III, p. 2578) about the futility of the 1924 La Follette Progressives' dream of replacing the Democrats as America's second party:
"First, the practical politics of the La Follette movement. Here in the East its supporters, the
New Republic among them, are arguing that the new party is to destroy and supplant the Democratic party as the opposition to conservative Republicanism. This seems to me impossible.
The Democratic party is more or less indestructible because of the solid South. [My emphasis--DT] A party which enters every campaign with roughly half the electoral votes [I assume that what Lippmann meant was "half of the electoral votes necessary for victory"--DT] is not in my opinion going to disappear. It seems extremely unlikely that La Follette will break the solid South, and almost as unlikely that the Southern Democrats will coalesce, as the
New Republic has suggested, with the Eastern Republicans. If the Democratic party survives, and if the Republican party survives, there is not under the presidential system of government any permanent future for a third party..."
There were plenty of voters in the South who were quite "radical" on economic issues and in that sense closer to La Follette than to Coolidge or Davis. But the tradition that only a white vote united behind the Democratic Party could preserve white supremacy in the South was hard to overcome.